On the use of non-linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues
On the use of non-linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues
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DOI：
10.1002/nme.2577
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发表时间：
2009-07-23
影响因子：
2.9
通讯作者：
Doblare, M.
中科院分区：
文献类型：
--
作者：
Alastrue, V.;Martinez, M. A.;Doblare, M.
基于微球的本构模型是模拟由定向排列的贡献元素组成的微结构的材料的有用工具。例如，纤维软组织就是这种情况。在这些模型中，宏观力学行为是通过将每个组分(例如每根纤维)在底层微球表面上的微观结构贡献进行积分来获得的，这使得可以将微观组分的力学特征结合到宏观响应中。这类模型和相关的数值技术的结合构成了一个强大的建模工具，因此需要一种有效的积分方案。在这方面，Bazant和oh(Zamm-J Appl Mech Mech Z Angew Math Mech 1986；66(1)：37-49)提出的单位球离散化被用来积分各向同性材料中的微观贡献。然而，各向异性的包含对于积分方案具有重要的影响，因为需要非常精细的离散化来准确地执行积分，这导致积分过程非常昂贵。此外，在微观结构层面使用内变量的本构模型需要在每个积分点的每个积分方向存储内变量，这使得这种方法相当复杂，对内存要求也很高。为了减少所需的积分方向数，本文提出了单位球面上旋转对称函数积分的几种非线性变换。对它们在积分冯·米塞斯取向分布函数中的精度进行了评价。此外，为了在非均匀变形问题的有限元背景下检验所提出的变换的精度和计算效率，采用了基于超弹性微球的软生物组织的本构关系。仿真结果表明，为了在合理的计算代价内准确地逼近积分的值，所提出的方法是合适的。版权所有(C)2009 John Wiley&Sons，Ltd.
Microsphere-based constitutive models are a helpful tool in the modelling of materials with a microstructure composed of contributing elements directionally arranged. This is the case, for instance, for fibred soft tissues. In these models, the macroscopic mechanical behaviour is obtained from the integration of the micro-structural contribution of each component (e.g. each fibre) over the surface of an underlying microsphere, which allows incorporating the mechanical features of the micro-constituents to the macroscopic response. The combination of this sort of models and the associated numerical techniques constitutes a powerful modelling tool for which an efficient integration scheme is required. In this regard, the unit sphere discretizations proposed by Bazant and Oh (ZAMM-J Appl Math Mech Z Angew Math Mech 1986; 66(1):37-49) have been used for the integration of the microscopic contributions in isotropic materials. Nevertheless, the inclusion of anisotropy has important implications with regard to the integration scheme, since very fine discretizations are needed to perform the integration accurately, causing the integration process to be very costly. In addition, the storage of internal variables at each integration direction of every integration point is required for constitutive models based on the use of internal variables at the micro-structural level, which renders this approach rather complex and memory demanding. In order to reduce the number of necessary integration directions, several non-linear transformations for the integration of rotationally symmetric functions over the Surface of the unit sphere are here presented. Their accuracy in the integration of the von Mises orientation distribution function is evaluated. Furthermore, a hyperelastic microsphere-based constitutive law for the modelling of soft biological tissues is used in order to check the accuracy and computational efficiency of the proposed transformations within a Finite Element context in inhomogeneous deformation problems. Simulation results show the suitability of the proposed methodology in order to accurately approximate the Value of the integrals within reasonable computational costs. Copyright (C) 2009 John Wiley & Sons, Ltd.